Poisson Distribution in Ideal Gas
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 19
3-9-2016
2829
Poisson Distribution in Ideal Gas
Consider a monatomic ideal gas of total
molecules in a volume
. Show that the probability, PN for the number N of molecules contained in a small element of V is given by the Poisson distribution
where
is the average number of molecules found in the volume V.
SOLUTION
The probability w1 of finding a particular molecule in a volume V is
The probability wn of finding N marked molecules in a volume V is
(1)
Similarly, the probability of finding one particular molecule outside of the volume V is
and 
for particular molecules outside V,
(2)
Therefore, the probability PN of finding any N molecules in a volume V is the product of the two probabilities (1) and (2) weighted by the number of combinations for such a configuration:
(3)
The condition 
also implies that 
Then we may approximate
(4)
So, (3) becomes
(5)
where we used the average number of molecules in V:
Noticing that, for large 
,
we obtain
(6)
where we used
(6) can be applied to find the mean square fluctuation in an ideal gas when the fluctuations are not necessarily small (i.e., it is possible to have (N - ⟨N⟩)/ ⟨N⟩~ 1, although N is always much smaller than the total number of particles in the gas 
).
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